Let $(M,g)$ be a (relatively compact) oriented (sub-)riemannian manifold without boundary. Let $d$ be the induced riemannian metric. I am looking for some constants $m,n,c,\epsilon>0$, such that for all $\delta\in(0,\epsilon)$ and $p\in M$ $$\int_M d(p,q)^m e^{-d(p,q)/\delta}dVol(q)\leq c\delta^{n+1}.$$
Is it also possible to get that type of approximation on an arbitrary metric measure space?
I would appreciate any idea how to tackle this approximation. This looks somehow like a gamma function, but since I never worked with them I was a bit lost in literature.
For any $m$, note
$$ e^{-x} = \frac{1}{e^x} \le \frac{1}{x^m/m!}=\frac{m!}{x^m},$$
we have
\begin{align*} \int_{M} d(p, q)^m e^{-d(p, q)/\delta} dV&\le m! \int_M d(p, q)^m \frac{\delta ^m }{d(p, q)^m} dV \le m! \operatorname{Vol}(M) \delta^m. \end{align*}
Obviously the above works whenever $(M, d)$ is a measure metric space with finite measure.